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CalculusTopics in calculus Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials | Mean value theorem | Vector calculus | Tensor calculus Differentiation Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates Integration Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals For other uses of the term calculus see calculus (disambiguation) Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas.Back to [[Software Update]]
One concept is differential calculus. It studies rates of change, which are usually illustrated by the slope of a line. Differential calculus is based on the problem of finding the instantaneous rate of change of one quantity relative to another. Examples of typical differential calculus problems are finding the following quantitiesThe Update Service:
The acceleration and speed of a free-falling body at * checks for updates to the application on a particular moment. background timerThe loss in speed and trajectory of * provides a fired projectile, such as an artillery shell or bullet. means for the user to check for updates to the applicationChange in profitability over time of * provide a growing business at a particular point in time. The other key concept is integral calculus. It studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. Examples set of integral calculus problems include finding the following quantities:controls for determining update behavior
Contents [hide]The system will automatically check for updates without user intervention:1 History * every 24 hours2 Differential calculus 3 Integral calculus 4 Foundations 5 Fundamental theorem of calculus 6 Applications 7 See also 8 Further reading 9 External links * on the first startup following an update, to check to see if the patch applied is the newest possible update or if there are newer ones.
[edit]HistoryMain article: History Regarding critical security updates, it would be useful to the user, if we could display a pop-up or similar sort (like what we have for-- " New updates available ") with the importance of calculusthis critical security update. This way we can inform the user the urgent need for the upgrade.
Though the origins of integral calculus are generally regarded as going no farther back than to the ancient Greeks, there is evidence that the ancient Egyptians may have harbored such knowledge as well. (See Moscow Mathematical Papyrus.) Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the area and volume of regions and solids. Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts. An Indian Mathematician, Bhaskara (1114-1185), gave an example of what is now called the "differential coefficient" and the basic idea of what is now known as "Rolle's theorem". = The 14th century Indian mathematician Madhava along with other mathematicians of the Kerala school made major inroads into Calculus that were not repeated anywhere in the world until the 17th century by Newton and Leibniz. Leibniz and Newton are usually designated the inventors of calculus, mainly for their separate discoveries of the fundamental theorem of calculus and work on notation.Update Check =
There has been considerable debate about whether Newton or Leibniz was first to come up with the important concepts of calculus. The truth of the matter will likely never be known. Leibniz' greatest contribution to calculus was his notation; he often spent days trying to come up with the appropriate symbol to represent a mathematical idea. This controversy between Leibniz and Newton was unfortunate in that it divided English-speaking mathematicians from those in Europe for many years, setting back British analysis (i.e. calculus-based mathematics) for a very long time. Newton's terminology and notation was clearly less flexible than that of Leibniz, yet it was retained in British usage until the early 19th century, when the work of the Analytical Society successfully saw the introduction of Leibniz's notation in Great Britain. It is now thought that Newton had discovered several ideas related to calculus earlier than Leibniz had; however, Leibniz was the first to publish. Today, both Leibniz and Newton # generate update service URL# determine if updates are considered to have discovered calculus independently.available# determine action# download patches# verify patches# install patches
Lesser credit for the development of calculus is given to Barrow, Descartes, de Fermat, Huygens, and Wallis. A Japanese mathematician, Kowa Seki, lived at the same time as Leibniz and Newton and also elaborated some of the fundamental principles of integral calculus, though this was not known in the West at the time, and he had no contact with Western scholars. [1]= Update Service URL =
[edit]The Service URL needs to incorporate data in these dimensions so as to reduce the complexity of the processing on the client side:Differential calculus* app nameMain article: Derivative* app locale* app version* app buildid (for distinguishing between nightlies on a "tester" build stream for example)* app buildtarget
The derivative measures the sensitivity of one variable to small changes in another variablee.g. Consider the formula:
for an object moving at constant speed<tt>/firefox/1. One's speed in a car describes the change in location relative to the change in time0. However, the speed itself may be changing and the formula above cannot account for that3. Calculus deals with this more complex but natural and familiar situation20050414/i586-pc-msvc/en-US/update.xml</tt>
Differential calculus determines the instantaneous speed, at any given specific instant in time, not just average speed during an interval of time. =The formula Speed Updates File= Distance/Time applied to a single instant is the meaningless quotient "zero divided by zero". This is avoided, however, because the quotient Distance/Time is not used for a single instant (as in a still photograph), but for intervals of time that are very short.
The derivative answers the question: as the elapsed time approaches zero, what does the average speed computed by Distance/Time approach? In mathematical language, this update.xml is an example of "taking a limitXML file that tells about available updates."It is formattedlike this:
More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient<tt><pre><?xml version="1.0"?>
The derivative of a function gives information about small pieces of its graph<updates> <update type="minor" version="1. It is directly relevant to finding the maxima and minima of a function — because at those points the graph is flat (i0.e4" extensionversion="1. the slope of the graph is zero)0"> <patch type="partial" url="http://www. Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the function by its tangent linesfoo. Differential calculus has been applied to many questions that are not first formulated in the language of calculuscom/1.0.4-partial.xpi" hashfunction="" hashvalue="" size=""/> <patch type="complete" url="http://www.foo.com/1.0.4-complete.xpi" hashfunction="" hashvalue="" size=""/> </update> .. <update type="major" version="1.1.2" extensionversion="1.1"> <patch type="complete" url="http://www.foo.com/1.1.2-complete.xpi" hashfunction="" hashvalue="" size=""/> </update></updates></pre></tt>
[edit]The <updates> list specifies the set of updates that can be downloaded andIntegral calculusMain article: Integralinstalled and may play a role in updating the application.
The definite integral evaluates Each "partial" update is a diff of the cumulative effect of many small changes new version from the previous version.If there are several "partial" updates available, they are all downloaded andinstalled in order. [Note: Initially we may only install a quantitysingle patch andthen rely on a subsequent update check to determine that there are more patchesavailable and install them at that time. The simplest instance is the formula]
for calculating the distance a car moves during Before a period collection of time when it updates is traveling at constant speed. The distance moved downloaded and installed, the size attributefor each patch is read to determine file size, and if the cumulative effect sum of the small distances moved in each patch sizes is found to be greater than the size of the many seconds the car "complete" patch (which is on a jar file whose contents are only file additions, removals and replaces, no file patches), then the road. The calculus "complete" file is able to deal with the natural situation in which the car moves with changing speeddownloaded.
Integral calculus determines the exact distance traveled during an interval We only supply "complete" updates to major versions since we cannot easily picka version to diff against off of time by creating a previous version series of better and better approximations, called Riemann sums, that approach e.g. do we diffoff 1.1.4? What if we do a security release 1.1.5 further down the line? Itis simpler to make users doing major upgrades redownload the exact distancebundle.
More formally, we say that This system intrinsically supports updates to the definite integral of updater - if a function on an interval point in timeis reached at which we can no longer fully update a limit user to the newest version,we can provide a series of Riemann sum approximationsupdates that take them to a version that can thenbe updated further, e.g.
Applications of integral calculus arise whenever the problem User is using 1.1.1Newest version is 1.5.9 but due to compute a number that is bug in principle (approximately) equal to the sum of updater in all versions older than 1.1.4, the solutions of many, many smaller problemsuser cannot update directly to 1.5.9.
Many of the functions that are integrated are rates, such as a speed. An integral of a rate of change of a quantity on an interval of time tells how much that quantity changes during that time period. It makes sense that if one knows their speed at every instant in time for an hour (i.e. they have an equation that relates their speed and time), then they should be able to figure out how far they go during that hour. The definite integral of their speed presents a method for doing so<tt><pre><?xml version="1.0"?>
Many of the functions that are integrated represent densities<updates> <update type="minor" version="1. If, for example, the pollution density along a river (tons per mile) is known in relation to the position, then the integral of that density can determine how much pollution there is in the whole length of the river1.2" extensionversion="1.1"> <patch type="partial" url="http://www.foo.com/1.1.2-partial.xpi" hashfunction="" hashvalue="" size=""/> <patch type="complete" url="http://www.foo.com/1.1.2-complete.xpi" hashfunction="" hashvalue="" size=""/> </update> <update type="minor" version="1.1.3" extensionversion="1.1"> <patch type="partial" url="http://www.foo.com/1.1.3-partial.xpi" hashfunction="" hashvalue="" size=""/> <patch type="complete" url="http://www.foo.com/1.1.3-complete.xpi" hashfunction="" hashvalue="" size=""/> </update> <update type="minor" version="1.1.4" extensionversion="1.1"> <patch type="partial" url="http://www.foo.com/1.1.4-partial.xpi" hashfunction="" hashvalue="" size=""/> <patch type="complete" url="http://www.foo.com/1.1.4-complete.xpi" hashfunction="" hashvalue="" size=""/> </update></updates>
Probability, the basis for statistics, provides one of the most important applications of integral calculus.</pre></tt>
[edit]FoundationsThe rigorous foundation of calculus is based on And for the notions of a function and of a limit; the latter has a theory ultimately depending on that of the real numbers as a continuum1. Its tools include techniques associated with elementary algebra, and mathematical induction1.4 user like so:
The modern study of the foundations of calculus is known as real analysis. This includes full definitions and proofs of the theorems of calculus. It also provides generalisations such as measure theory and distribution theory<tt><pre><?xml version="1.0"?>
[edit]<updates>Fundamental theorem of calculus <update type="minor" version="1.1.5" extensionversion="1.1"> <patch type="partial" url="http://www.foo.com/1.1.5-partial.xpi" hashfunction="" hashvalue="" size=""/> <patch type="complete" url="http://www.foo.com/1.1.5-complete.xpi" hashfunction="" hashvalue="" size=""/> </update> <update type="major" version="1.5.9" extensionversion="1.5">The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations <patch type="complete" url="http://www.foo.com/1. More precisely, antiderivatives can be calculated with definite integrals, and vice versa5.9-complete.xpi" hashfunction="" hashvalue="" size=""/> </update></updates>
This connection allows us to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter.</pre></tt>
This realizationSo the user of 1.1.1 will have the 1.1.2, 1.1.3, made by both Newton and Leibniz1.1.4 patchesdownloaded and applied in that order. When they start the application the nexttime, the application will recheck for updates using 1.1.4's enhanced bugfixedupdater, was key to and discover 1.1.5 and the massive proliferation of analytic results after their work became known1.5.9 major update.
The fundamental theorem provides an algebraic method This implies that the database that manages all of computing many definite integrals --without performing limit processes--by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function this version informationhas to know that some updates can only apply to its derivatives, and are ubiquitous in the sciencescertain version (ranges).
1st Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and F is an antiderivative of f on the interval [a, b], then
[edit]ApplicationsThe development = Preference Controls and use of calculus has had wide reaching effects on nearly all areas of modern living. It underlies nearly all of the sciences, especially physics. Virtually all modern developments such as building techniques, aviation, and other technologies make fundamental use of calculus. Many algebraic formulas now used for ballistics, heating and cooling, and other practical sciences were worked out through the use of calculus. In a handbook, an algebraic formula based on calculus methods may be applied without knowing its origins.State =
The success * <tt>app.update.enabled</tt> <blockquote> <table border="1" cellspacing="0" cellpadding="3"> <tr><td><tt>'''true'''</tt></td><td>Enables background update checking</td></tr> <tr><td><tt>'''false'''</tt></td><td>Disables background update checking</td></tr> </table> </blockquote>* <tt>app.update.mode</tt> <blockquote> <table border="1" cellspacing="0" cellpadding="3"> <tr><td>'''0'''</td> <td>automatically download updates for minor and major updates, regardless of calculus has been extended over time to differential equations, vector calculusincompatibilities that may arise with addons</td></tr> <tr><td>'''1'''</td> <td>automatically download updates for minor and major releases, calculus of variationsif no incompatibilities with addons are present, complex analysisotherwise prompt</td></tr> <tr><td>'''2'''</td> <td>automatically download updates for minor releases, prompt about major releases</td></tr> <tr><td>'''3'''</td> <td>prompt about minor and differential topologymajor releases</td></tr> </table> </blockquote>* <tt>app.update.interval</tt> <blockquote> <table border="1" cellspacing="0" cellpadding="3"> <tr><td>'''86400'''</td><td>seconds between update checks</td></tr> </table> </blockquote>* <tt>app.update.timer</tt> <blockquote> <table border="1" cellspacing="0" cellpadding="3"> <tr><td>'''5000'''</td><td>milliseconds between app.update.interval expiry checks</td></tr> </table> </blockquote>* <tt>app.update.silent</tt> <blockquote> <table border="1" cellspacing="0" cellpadding="3"> <tr><td>'''true'''</td><td>all update prompting should be suppressed</td></tr> <tr><td>'''false'''</td><td>show prompts to the user when there are events they should respond to</td></tr> </table> </blockquote>* <tt>app.update.lastUpdateDate.background-update-timer</tt> <blockquote> <table border="1" cellspacing="0" cellpadding="3"> <tr><td>'''1114648397'''</td><td>seconds since epoch of last update time</td></tr> </table> </blockquote>* <tt>app.update.url</tt> <blockquote> <table border="1" cellspacing="0" cellpadding="3"> <tr><td>'''http://aus.mozilla.org/update/firefox/%1%/%2%/%3%/update.xml'''</td></tr> <tr><td>%1% is the version, in FVF containing the build id; %2% is the build target (OS+Architecture); %3% is the ab-CD locale</td></tr> </table> </blockquote> Back to [[Software Update]]
Reverted edit of Georgebush, changed back to last version by Ben
= The amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure. The amount of money accumulated by a business under varying business conditions. The amount of parking lot plowed by a snowplow of given power with varying rates of snowfall. The two concepts, differentiation and integration, define inverse operations in a sense made precise by the fundamental theorem of calculus. In teaching calculus, either concept may be given priority. The usual educational approach is to introduce differential calculus first.Background Timer =
The derivative lies at application should provide a preference setting that can be set to holdthe heart of the physical sciencesapplication within one version range, e.g. within 1. Newton's law of motion, Force = Mass × Acceleration, has meaning in calculus because acceleration is a derivative0. Maxwell's theory of electromagnetism and Einstein's theory of gravity (general relativity) are also expressed in the language of differential calculusx, as is never updatingto the basic theory of electrical circuits and much of engineeringnewest major version but only installing incremental security updates.
The classic geometric application is to area computationsupdate. In principle, xml file for the area of a region can be approximated by chopping it up into many very tiny squares and adding the areas of those squares. (If the region has a curved boundary, then omitting the squares overlapping the edge does not cause too great an error1.) Surface areas and volumes can also be expressed as definite integrals1.1 user might look something like this:
On the client side: We can have a [Update Details] button under menu: [Help] > [About Mozilla Firefox].2nd Fundamental Theorem of Calculus: If f is continuous on an open interval I containing By clicking this button [Update Details] we can present a, then, for every x in tabluar view with the interval,following columns: 1. Updated Item 2. Version 3. Type { Security update or enhancement update...} 4. Updated Item release date 5. Updated date 6. Updated by user ?
